Name: Introduction to Inverse of Matrices
Uploaded: 2019-07-19
Duration: 11 min 27 s
Description: In this video, we will study the inverse of matrices and conditions for the existence of inverses and invertible matrices, along with some of their properties.

Question

If the inverse of a square matrix exists, prove that it is unique. If

A

and

B

are both invertible square matrices of the same order, prove that

(A B)^{- 1} = B^{- 1} A^{- 1}

.

Accepted Answer

Let $A$ be an invertible matrix of order $n \times n$ . Let $B$ and $C$ be two inverses of $A$ . Then,

$A B = B A = I_{n} . . . (1)$

$A C = C A = I_{n} . . . (2)$

Now, $A B = I_{n}$

$\Rightarrow C (A B) = C I_{n}$ [Pre-multiplying both sides by $C$ ]

$\Rightarrow (C A) B = C I_{n}$

$\Rightarrow I_{n} B = C I_{n} [∵ C A = I_{n}]$

$\Rightarrow B = C [∵ I_{n} B = B and C I_{n} = C]$

Hence, an invertible matrix possesses a unique inverse.

It is given that $A$ and $B$ are invertible matrices.

$∴ | A | \neq 0 and | B | \neq 0$

$\Rightarrow | A | | B | \neq 0$

$\Rightarrow | A B | \neq 0$

$\Rightarrow A B$ is an invertible matrix. $[∵ | A B | = | A | | B |]$

Now, $(A B) (B^{- 1} A^{- 1}) = A (B B^{- 1}) A^{- 1}$

$\Rightarrow (A B) (B^{- 1} A^{- 1}) = (A I_{n}) A^{- 1} [∵ B B^{- 1} = I_{n}]$

$\Rightarrow (A B) (B^{- 1} A^{- 1}) = A A^{- 1} [∵ A I_{n} = A]$

$\Rightarrow (A B) (B^{- 1} A^{- 1}) = I_{n} [∵ A A^{- 1} = I_{n}]$

and $(B^{- 1} A^{- 1}) (A B) = B^{- 1} (A^{- 1} A) B$

$\Rightarrow (B^{- 1} A^{- 1}) (A B) = B^{- 1} (I_{n} B) [∵ A^{- 1} A = I_{n}]$

$\Rightarrow (B^{- 1} A^{- 1}) (A B) = B^{- 1} B [∵ I_{n} B = B]$

$\Rightarrow (B^{- 1} A^{- 1}) (A B) = I_{n} [∵ B^{- 1} B = I_{n}]$

Thus, $(A B) (B^{- 1} A^{- 1}) = I_{n} = (B^{- 1} A^{- 1}) (A B)$

Hence, $(A B)^{- 1} = B^{- 1} A^{- 1}$ .

If the inverse of a square matrix exists, prove that it is unique. If $A$ and $B$ are both invertible square matrices of the same order, prove that $(A B)^{- 1} = B^{- 1} A^{- 1}$ .

Important Questions on Matrices

Learn and Prepare for any exam you want !

If the inverse of a square matrix exists, prove that it is unique. If A and B are both invertible square matrices of the same order, prove that (AB)−1=B−1A−1.

Important Questions on Matrices

Learn and Prepare for any exam you want !

If the inverse of a square matrix exists, prove that it is unique. If $A$ and $B$ are both invertible square matrices of the same order, prove that $(A B)^{- 1} = B^{- 1} A^{- 1}$ .